-3
$\begingroup$

While preparing a tutorial session for my students, I come across a very challenging integration, which is :$$ \int_{-\infty}^\infty e^{t^2 }\ \mathrm{d}t$$

I attempted many methods to solve it but with no use. Your assistance will be greatly appreciated.

$\endgroup$
5
  • $\begingroup$ Can you do $\int t e^{t^2} dt$. $\endgroup$ – Z Ahmed Jan 16 at 16:01
  • 2
    $\begingroup$ Depends on how you interpret $\int$. If it means $\int_{-\infty}^\infty$, then this is a well-known result. If you looking for an antiderivative, then your tutorial session is not good... $\endgroup$ – user9464 Jan 16 at 16:05
  • $\begingroup$ Thank you for our response . well it is integration from - infinity to + infinity ! what is the well-know result you referred to ? by the way it is tutorial on linear system analysis $\endgroup$ – Safa Jan 16 at 16:11
  • $\begingroup$ Then you should add those bounds in your integral. See en.wikipedia.org/wiki/… $\endgroup$ – user9464 Jan 16 at 16:35
  • $\begingroup$ Great ! This is what I was looking for , thanks alot $\endgroup$ – Safa Jan 16 at 17:15
1
$\begingroup$

Assuming you are looking for the antiderivative of $e^{t^{2}}$: $$\int e^{t^2 }\ \mathrm{d}t = \frac{\sqrt{\pi}}{2}\operatorname{erfi}(x),$$where $\operatorname{erfi}(x)$ is the imaginary error function defined as $$\operatorname{erfi}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{t^{2}}\mathrm{d}t.$$ There is no way of expressing the antiderivative of $e^{t^{2}}$ in terms of elementary functions.

$\endgroup$
6
  • $\begingroup$ Brilliant !!! wow you are a calculus genius . Thank you so much . $\endgroup$ – Safa Jan 16 at 16:17
  • $\begingroup$ From where can learn such advance calculus ? I should be grateful if you would recommend a resource to me . $\endgroup$ – Safa Jan 16 at 16:18
  • $\begingroup$ Honestly the wikipedia page is a pretty good start if you just want some of the terminology. Error functions are just one of many types of special functions. $\endgroup$ – DMcMor Jan 16 at 16:20
  • $\begingroup$ luckily I found a video on YouTube that solves the same problem .But he uses erf(x) function instead ! youtube.com/watch?v=jkytxdedxhU $\endgroup$ – Safa Jan 16 at 16:47
  • 1
    $\begingroup$ Well noted . many thanks $\endgroup$ – Safa Jan 16 at 19:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.